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Abstract

. We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S r d (4) with d 3r+2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the the smallest angle in the underlying triangulation and the nature of the boundary of the domain. AMS(MOS) Subject Classifications: 41A15, 41A63, 41A25, 65D10 Keywords and phrases: Bivariate Splines, Approximation Order by Splines, Stable Approximation Schemes, Super Splines. x1. Introduction Let\Omega be a bounded polygonal domain in IR 2 . Given a finite triangulation 4 of \Omega\Gamma we are interested in spaces of splines of smoothness r and degree d of the form S r d (4) := fs 2 C r(\Omega\Gamma : sj T 2 P d ; for all T 2 4g; where P d denotes the space of polynomials of...